1. **State the problem:** Simplify the expression $$\frac{x^2 - 25}{x^2 + 5x} \div \frac{xy + 6x - 5y - 30}{5x - 15}$$. 2. **Rewrite division as multiplication:** Dividing by a fraction is the same as multiplying by its reciprocal. $$\frac{x^2 - 25}{x^2 + 5x} \times \frac{5x - 15}{xy + 6x - 5y - 30}$$ 3. **Factor all polynomials:** - Numerator top-left: $$x^2 - 25 = (x - 5)(x + 5)$$ (difference of squares) - Denominator top-left: $$x^2 + 5x = x(x + 5)$$ - Numerator top-right (denominator of division): $$5x - 15 = 5(x - 3)$$ - Denominator top-right (numerator of division): Factor by grouping: $$xy + 6x - 5y - 30 = x(y + 6) - 5(y + 6) = (x - 5)(y + 6)$$ 4. **Rewrite expression with factors:** $$\frac{(x - 5)(x + 5)}{x(x + 5)} \times \frac{5(x - 3)}{(x - 5)(y + 6)}$$ 5. **Cancel common factors:** - Cancel $(x + 5)$ from numerator and denominator. - Cancel $(x - 5)$ from numerator and denominator. Intermediate step showing cancellation: $$\frac{\cancel{(x - 5)}(x + 5)}{x\cancel{(x + 5)}} \times \frac{5(x - 3)}{\cancel{(x - 5)}(y + 6)} = \frac{1}{x} \times \frac{5(x - 3)}{y + 6}$$ 6. **Multiply remaining factors:** $$\frac{1}{x} \times \frac{5(x - 3)}{y + 6} = \frac{5(x - 3)}{x(y + 6)}$$ **Final simplified expression:** $$\boxed{\frac{5(x - 3)}{x(y + 6)}}$$